The Number Line

Introduction

The number line is an illustration tool used to visualize many of the concepts in mathematics. It allows us to 'see' the numbers we work with, allowing us to:

  1. 'see' which values are greater than and less than others
  2. better comprehend the difference between quantities by using scale and distance
  3. identify trends between sets of numbers
  4. visualize the fundamental operations of arithmetic and beyond.

The number line is the third way we have to describe the domain of a set. The first two are called set-builder and interval notation, go to Set Theory: Interval Notation to learn more.

The Coordinate

The first concept to cover is that of a coordinate, or point. A coordinate is how we illustrate the position of a single value. To illustrate the position of zero, first we place a dot, then place the number zero next to it.

coordinate of zero

Before placing the next point, we must introduce the concept of relationship. If a point is placed to the right of another point, then it is greater than the point on its left. Likewise, if a point is placed to the left of another, then it is less than the point to its right.

The point zero is called the origin and it is to the relation to this point that the sign of a number is determined. Any point placed to the right of the origin is positive. Any point placed to the left of the origin is negative.

sign with relation to the origin

Zero is generally defined as the value representing nothing, or as a starting point when travelling. Any value greater than zero (or any distance travelled in a specified direction) is named positive. A negative value describes a value less than zero (or distance travelled in the opposite direction).

When placing a positive one into the illustration, it goes to the right of zero like so:

coordinates zero and one

Scale

When we place more than two points into the illustration, then we must establish a scale. The scale is the amount of distance that we determine a specified quantity to be. Once established, this distance will be used to quantify the distance between each pair of points.

This reference distance has already been established in our illustration. The distance between the origin and the coordinate 1 (the first point we placed) is exactly one unit (a unit being any mode of measure). Now if we want to add the coordinate for the value 3, we would place it three times the distance of the first point (or three units) from the origin to the right.

coordinates zero and one

If we wanted to add the coordinate for -1, we would place it the same distance from the origin as the coordinate for 1, but to the left of the origin (negative value).

coordinates zero and one

The Number Line

There are an infinite number of: whole numbers, negative integers, and fractions between one and zero. There are an infinite number of numbers. To represent this vast array of values, we use a line.

line

The solid line represents the infinite number of dots we would have place side-by-side to represent the infinite number of values within a given range. The arrows represent the numbers which stretch into infinity. The Real Number Line is the line which represents every number within the real number set.

the real number line

Range

The range of a set is a description of the values it contains. To visualize a set containing many individuals points, mark each point with a dot (or a filled in circle) on the real number line.

illustration for the set A={1,2,5,8}

To illustrate a set containing all values between two points, we draw thick solid line connecting the two points.

illustration for the set A=[5,9]

We illustrate a range which extends to infinity in this fashion with a bolded arrow.

illustration for the set A=(-inf,7]

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